When Scar took over the Pride Lands, he was the only Adult Male Lion in the Kingdom - At least, the only one at Pride Rock of the age to mate. In one essay I have read, it was suggested that Scar's reign lasted five years, for it takes five years for a lion's mane to fully grow (so from when he exiled Simba to Simba's return). By the time of the young prince's exile, there were at least ten females capable of mating within the Pride, and only one AML was present during that time - Scar.
During the log scene in Hakuna Matata, we see Simba in three stages - Juvenile, Adolescent, and Adult. If we reckon that adolescent is midway between juvenile and adult, and that Simba was in the jungle for five years, then we could estimate that it takes approximately two years and a half to grow to the adolescent stage. By the time Kiara had become a juvenile in Simba's Pride, Nuka is clearly an adolescent (if not adult), which makes him at the least two and a half years old. Kovu and Vitani, however, are still juveniles.
Considering that between TLK and SP, Simba and Nala's cub from the first film disappears, and both of their bodies become more adult (Simba's mane thickens, Nala's build becomes similar to Sarabi's; more mature and less lithe), it is plausible that at least a couple of years have passed. This means that Nuka had to of been born before Simba overthrew Scar, and seeing as there are no other AMLs in the Pride Lands during his reign, this means that Nuka had to be Scar's cub.
This is also hinted at several times in SP, for instance, with Nuka saying "Kovu, Kovu, Kovu. Scar wasn't even his father." This bitterness at Kovu being the heir instead of him, especially when judging Nuka's words, make it sound like he is in fact Scar's son - Which, as we've just proven, he is.
Now, consider the picture to the right - Simba, Kopa, and Kovu all havecircles in their inner ears. When Simba is an adolescent, he still bears his ear tufts. However, when Nuka is first seen in SP, and when Taka/Scar is depicted in A Tale of Two Brothers, neither of the lions have the ear circles.
Neither Nala, Kiara, or Vitani (none of the girl cubs - None of the females, apart from Sarabi) have the ear rings, which I think makes it something hereditary that crops up most in male genes rather than female. The fact that Sarabi has them, that Simba has them, and that Kopa has them, make me think it IS hereditary to a certain exent, which could suggest who Kovu's father is.
Now, for the famous 'dark fur' arguement. While it is suggested by some that the reason for Kovu's reddish-brown pelage is due to Scar being his father, it is also possible that Kovu's fur colour is inherited from Mohatu, Simba's grandfather. However, there is still one remaining problem - Kovu's nose.
This is one of the major flaws in my theory. The only lions that have been (officially) depicted with the 'M' nose are: Scar, Kovu, Vitani, and Nuka. To my knowledge, not one lion apart from them (even including official comics) bare the same nose. In an early draft of SP, Scar is Kovu's father, which would make a lot of sense, but the latter's appearance still may be explained away that the nose, like my theory about fur colouring, is something inherited from a later generation - Perhaps from Uru, Ahadi's mate, who has not been depicted in any of the official films or comics (or text, although I am do not know for sure on this point).
This is the thorn in the paw, or so the saying goes. In SP, it is repeteadly stated that Scar is NOT Kovu's father, despite looking quite similar to him (his muzzle and mane are different, and the latter feline's fur is more reddish-brown than reddish-orange). However, all three of Zira's cubs bare similarities to Scar in some way (although a few aspects could be attributed to Zira herself): Vitani bares Scar's long and toothy muzle; Kovu has similar fur shades and the same eye colouring; and Nuka has the same malnourished, thin body. And lastly, all of her cubs bare the 'M' nose.
One must remember, of course, that the decision to change Scar from being Kovu's father was only made by Disney due to Kovu's relationship with Kiara if it was not changed would be the thing they fear so much: Incest! Also, another added complication to the equation is the factor that whilst (on my formentioned scale of being a couple of years between Kopa and Kiara's births) it would mean that Scar couldn't of been Vitani or Kovu's father, Zira DOES state in the film that Kovu was 'Hand-picked by Scar'. This could lead to the possible explanation that Kovu is older than Kiara, and that the gap between Simba's cubs being born is shorter than I estimated. If so, that would mean that (since it's highly unlikely he would allow a rogue to impregnate one of his Pride, although it IS Zira we're talking about here, she could of done so anyway) Simba would be Vitani and Kovu's father.
I conclude with this note - Scar is clearly Nuka's father, Vitani and Kovu should be Simba's cubs if not for a problem with wording in the second film, or that they were Scar's anyhow devoid of other matter.
Notes: Officially, the cub that is presented at the end of The Lion King is Kiara and the ceremony is the same one that features at the start of Simba's Pride. The difference of the ceremonies and appearance of cubs is most probably due to the artists from the first film not being aware that a second would be made, and some changes by the management team for Simba's Pride, including the change of Simba's cub to a girl. Kovu, Vitani, and Nuka are all Zira's cubs. Their official father is unknown.